LAWS OF MOTION Aristotle’s Fallacy and Galileo’s Observations

LAWS OF MOTION Aristotle’s Fallacy and Galileo’s Observations Newton’s First Law of Motion – Inertia: Eg. of Inertia of Rest and Inertia of Motion Momentum Newton’s Second Law of Motion and Derivation of F = ma Some Important Points about Second Law Impulse – Importance of Impulse in Practice Newton’s Third Law of Motion – Applications / Examples of Third Law Horse and Cart Problem Law of Conservation of Linear Momentum – Scalar & Vector Treatment Equilibrium of a Particle Common Forces in Mechanics Friction – Static, Kinetic and Rolling Friction – Friction as a Necessary Evil Methods of Reducing Friction Circular Motion – Bending of a Cyclist Car on a Level Circular Road and Car on a Banked Circular Road Solving Problems in Mechanics Created by C. Mani, Assistant Commissioner, KVS RO Silchar Next

Galileo’s observations
Aristotle’s Fallacy Aristotelian law of motion states that “an external force is required to keep a body in motion”. It was proved wrong by Galileo Galilei and Isaac Newton. Galileo’s observations A ball released from rest on one plane rolls down and climbs up the other. In the absence of friction, the heights of the ball on both the planes are the same (ideal condition). In practical situation, the height on the second plane is always a little less but never greater. Home Next Previous

Case-1 Case-2 Case-3 A ball released from rest in case-1 reaches the same height on the other side (in the absence of friction). In case-2, the ball reaches the same height on the other side (in the absence of friction) but travels longer distance. In case-3, which is the limiting case, the ball rolls down and continues to move in the horizontal direction through infinite distance (in the absence of friction). Home Next Previous

Galileo’s Law of Inertia:
Galileo arrived at a new insight that “the state of rest and the state of uniform linear motion are equivalent”. In both the cases, there is no net force acting on the body. Therefore, there is no need to apply an external force to keep a body in uniform motion. We need to apply external force only to encounter the frictional or any other force, so that the forces sum up to zero net external force. To summarise, if the net external force is zero, a body continues to remain at rest and a body in uniform motion continues to move with uniform velocity. This property of the body is called ‘inertia’. Inertia means ‘resistance to change’. Galileo’s Law of Inertia: A body does not change its state of rest or of uniform motion, unless an external force compels to change its state. Home Next Previous

NEWTON’S FIRST LAW OF MOTION
Every body continues to be in its state of rest or of uniform motion in a straight line unless and until compelled by an external force to change its state of rest or of uniform motion. In simple words, if the net external force on a body is zero, its acceleration is zero. Or, acceleration can be non-zero only if there is a net external force on the body. Reaction (R) (Exerted by the table) Since the block is at rest, the normal reaction R must be equal and opposite to its weight W. Weight (W) (Force of gravity) Home Next Previous

The block remains at rest…..
F The block remains at rest….. unless and until it is acted upon by an external force. F The ball continues to be in uniform motion…… unless and until it is acted upon by an external force. The car continues to be in uniform motion…… in the absence of an external force. Home Next Previous

Inertia Inertia of rest
Inertia is the inherent property of a body due to which it resists a change in its state of rest or of uniform motion. Inertia can be understood in parts, viz. inertia of rest, inertia of motion and inertia of direction. Mass is a measure of the inertia of a body. Heavier objects have more inertia than lighter objects. Eg. 1. A stone of size of a football has more inertia than football. 2. A cricket ball has more inertia than a rubber ball of the same size. Inertia of rest Home Next Previous

Examples of Inertia of rest:
A passenger in a bus jerks backward when the bus starts suddenly because the passenger tends to be in inertia of rest whereas the bus is moved away forcefully. When a bed sheet is flicked away suddenly dust particles fall away as they tend to be in inertia of rest. When a branch of a tree carrying a mango is suddenly flicked mango falls off due to inertia of rest. Home Next Previous

Examples of inertia of motion:
A passenger in a bus jerks forward when the bus stops suddenly because the passenger tends to be in inertia of motion whereas the bus is stopped forcefully. An athlete after reaching the finishing point can not stop suddenly or if he stops suddenly then he falls toppling head down. A car takes some time and moves through some more distance before coming to rest even after the application of brakes. A rotating fan continues to do so for some more time even after the current is switched off. An oscillating simple pendulum bob does not halt at the mean position but continues to move further. When a car or bus turns around a sharp corner, we tend to fall sideways because of our inertia to continue to move in a straight line. It is dangerous to jump out of a moving bus because the jumping man’s body is in the state of inertia of motion but the legs are suddenly stopped by the ground and hence he topples down. Home Next Previous

MOMENTUM Momentum is the quantity of motion in a body and it depends on its mass and velocity. Momentum of a body is defined as the product of its mass and velocity. i.e. Momentum = mass x velocity or p = m x v Momentum is directly proportional to mass. If a cricket ball and a tennis ball move with same velocity, momentum of cricket ball is more because its mass is larger than that of the tennis ball. Momentum is directly proportional to velocity. If two cricket balls move with different velocities, then the momentum of the ball with greater velocity possesses more momentum. If a body is at rest, its velocity is zero and hence its momentum is zero. But, every moving body possesses momentum. Momentum is a vector quantity. SI unit of momentum is kg m/s or kg ms-1. CGS unit of momentum is g cm/s or g cms-1. Home Next Previous

Importance of Momentum for considering the effect of force on motion:
Eg. Suppose a light vehicle (say a car) and a heavy vehicle (say a truck) are parked on the horizontal road A greater force is required to push the truck than the car to bring them to the same speed in same time Similarly, a greater opposing force is required to stop the truck than the car, if they are moving with the same speed. A bullet fired by a gun can easily pierce human tissue before it stops, resulting in casualty The same bullet with moderate speed will not cause much damage Thus, for a given mass, the greater the speed, the greater is the opposing force needed to stop the body in a certain time. The same force for the same time causes the same change in momentum in different bodies. The force not only depends on the change in momentum, but also on how fast the change is brought. The greater the rate of change of momentum, the greater is the force. Home Next Previous

The above examples do not indicate that momentum is a vector.
The following example shows that momentum is a vector. Suppose a stone is whirled with uniform speed in a horizontal circle by means of a string. The magnitude of momentum is constant but its direction keeps changing continuously. A force is needed to change this momentum vector. This force is provided by hand through the string and it is also felt by the hand. Home Next Previous

NEWTON’S SECOND LAW OF MOTION
The rate of change of momentum of a body is directly proportional to the applied force and takes place in the direction in which the force acts. Force α Time taken Change in momentum If force F changes the velocity of a body of mass m from v to v + Δv, then its initial momentum p = mv will change by Δp = m Δv. According to Newton’s Second Law, dt (mv) F = k d F α Δp Δt dt F = k m dv F = k Δp Δt or F = k m a where k is a constant of proportionality. If force of 1 N applied on a body of mass 1 kg produces an acceleration of 1 m/s2 on the body, then Δp Δt lim Δt→0 F = k 1 = k x 1 x 1 or k = 1 or F = m a dp dt F = k or Force = mass x acceleration Home Next Previous

Force is a vector quantity.
Acceleration a = F m The acceleration produced in a body is directly proportional to the force acting on it and inversely proportional to the mass of the body. Force is a vector quantity. Force can cause acceleration or deceleration. Eg.: Accelerator of a car accelerates it and brakes decelerate it. SI unit of force is ‘newton’. One ‘newton’ is defined as that force which when acting on a body of mass 1 kg produces an acceleration of 1 m/s2 in it. Place an 100 g on your outstretched palm. The force you feel is nearly 1 newton ! 100 g Home Next Previous

Some important points about Second Law
In the Second Law, F = 0 implies a = 0. This is obviously consistent with the First Law. The Second Law is a vector law. It is equivalent to three equations, one for each component of the vectors: dpx dt Fx = = max dpy dt Fy = = may dpz dt Fz = = maz This means that if a force is not parallel to the velocity of the body, but makes some angle with it, it changes only the component of velocity along the direction of force. The component of velocity normal to the force remains unchanged. For e.g. in the motion of a projectile under the vertical gravitational force, the horizontal component of velocity remains unchanged. Home Next Previous

IMPULSE 3. The equation dp dt F =
is applicable to a single point particle. = m a However, it is same for a rigid body or, even more generally, to a system of particles. In this case, m refers to the mass of rigid body or total mass of the system of the particles and a refers to the acceleration of the centre of mass of the system. Any internal forces in the system are not to be included in F. The Second Law is a local relation which means that force F at a point in space at a certain instant of time is related to a at that point at that instant. Acceleration here and now is determined by the force here and now, not by any history of the motion of the particle. IMPULSE When a ball hits a wall and bounces back, the force on the ball by the wall acts for a very short time when the two are in contact, yet the force is large enough to reverse the momentum on the ball. In these situations, it is difficult to ascertain the force and time duration separately. However, the product of force and time, which is the change in momentum is a measurable quantity. This product is called ‘impulse’. Home Next Previous

Importance of Impulse in Practice
Impulse = Force x time duration = Change in momentum j = F t A large force acting for a short time to produce change in momentum is called ‘impulsive force’. Importance of Impulse in Practice A high jumping or long jumping athlete is provided either a cushion or a heap of sand on the ground to fall upon. The cushion or sand helps to increase the time in which the momentum comes to zero. This decreases the rate of change of momentum and hence the force. So, the athlete does not get hurt. Packing materials like thermocoal, corrugated sheets, bubbled plastic sheet, straw, paper strands, etc. are used while packing glassware, chinaware, electronic devices, etc. These materials help to increase the time in which the momentum comes to zero when jolting and jerking take place. This decreases the rate of change of momentum and hence the force. So, the articles do not get broken. Home Next Previous

Shock absorbers are provided in vehicles to allow more time during the bumps and jerks.
A seasoned cricket player stops the ball (catches) by drawing his hands along the direction of motion of the ball thus by allowing more time and to apply less force. A fast moving cricket ball has a large momentum In stopping the cricket ball, its momentum has to be reduced to zero When a player moves his hands back, the time taken to stop the ball increases and hence the rate of change of momentum decreases i.e. the force exerted by the ball on the hands decreases So, the hands of the player do not get hurt. Home Next Previous

NEWTON’S THIRD LAW OF MOTION
To every action, there is always an equal and opposite reaction. The statement means that in every interaction, there is a pair of forces acting on the two interacting objects. The size of the forces on the first object equals the size of the force on the second object. The direction of the force on the first object is opposite to the direction of the force on the second object. Forces always come in pairs - equal and opposite (action-reaction) force pairs. FAB = - FBA Note: Action and reaction are just forces. The forces always occur in pairs. Action and reaction do not act on the same body. Action and reaction act on different bodies but simultaneously. There is no cause-effect relation as it is a misconception. Though action and reaction forces are equal in magnitude but they do not produce equal acceleration in the two bodies (of different masses) on which they act. Home Next Previous

Examples / Applications of Newton’s Third Law of motion
Reaction = 58 gwt Action = 58 gwt 20 40 80 60 100 20 40 80 60 100 Home Next Previous

F Action F Reaction Home Next Previous

Recoil of a gun F F Force on bullet (Action) Force on gun (Reaction)
Home Next Previous

Action and Reaction Home Next Previous

Vertical component of reaction
Forward Motion Vertical component of reaction Reaction Reaction Horizontal component of reaction Action Weight Weight Home Next Previous

Action Reaction Home Next Previous

Horse and Cart Problem A horse is urged to pull a cart. The horse refuses to try, citing Newton’s third law as his defence. “The pull of me on the cart is equal but opposite to the pull of the cart on me. If I can never exert a greater force (action and reaction are always equal) on the cart than it exerts on me, how can I ever set the cart moving?”, asks the horse. How would you reply? Home Next Previous

Why don’t you educate the Horse properly?
RC TCH THC V Reaction R f H WC Action WH Why don’t you educate the Horse properly? Reaction ‘RC’ on the cart offered by the ground must be equal and opposite to weight of the cart ‘WC’. Reaction pull of the cart on the horse ‘TCH’ must be equal and opposite to forward pull of the horse on the cart ‘THC’. If the horse pushes the ground in a slanting manner (Action), the Reaction offered by the ground is resolved into Vertical and Horizontal components. The Vertical component ‘V’ must be equal and opposite to weight of the horse ‘WH’. If the Horizontal component ‘H’ is greater than the Friction ‘f’, then the horse-cart system will move forward with acceleration. Home Next Previous

LAW OF CONSERVATION OF MOMENTUM
The total momentum of an isolated system of interacting particles is conserved. OR When two or more bodies act upon one another, their total momentum remains constant provided no external forces are acting on them. Home Next Previous

Scalar Treatment Suppose a big and a small car move in the same direction with different velocities. Let the mass of the bigger car be ‘m1’ and its initial velocity is ‘u1’. Let the mass of the smaller car be ‘m2’ and its initial velocity is ‘u2’ such that u2 < u1. Suppose both the cars collide for a short time ‘t’. Due to the collision, the velocities will change. Let the velocities after the collision be v1 and v2 respectively. Home Next Previous

According to Newton’s third law, F21 = – F12
Suppose that during collision, the bigger car exerts a force F21 on the smaller car and in turn, the smaller car exerts a force F12 on the bigger one. When the force F21 acts on the smaller car, its velocity changes from u2 to v2. F21 = m2a2 F21 = m2 v2 – u2 t When the force F12 acts on the bigger car, its velocity changes from u1 to v1. F12 = m1a1 F12 = m1 v1 – u1 t According to Newton’s third law, F21 = – F12 m2 v2 – u2 t m1 v1 – u1 = - Cancelling t on both sides, we get m2(v2 – u2) = m1(v1 – u1) m2v2 – m2u2 = m1v1 – m1u1 Initial momentum of the bigger car = m1u1 Initial momentum of the smaller car = m2u2 Final momentum of the bigger car = m1v1 Final momentum of the smaller car = m2v2 m1v1 + m2v2 = m1u1 + m2u2 Total momentum after collision = Total momentum before collision Home Next Previous

Vector Treatment vA vB vA’ vB’ FBA FAB mA mB
Click to see the collision… Consider two bodies A and B of mass mA and mB moving with velocities vA and vB respectively. Their initial momenta are pA and pB. Let the bodies collide, get apart and move with final momenta pA’ and pB’ respectively. By Newton’s Second Law, FAB Δt = pA’ - pA and FBA Δt = pB’ - pB where Δt is the common interval of time for which the bodies are in contact. By Newton’s Third Law, FAB = - FBA or FAB Δt = - FBA Δt or pA’ - pA pB’ - pB) = - ( i.e. pA’ + pB’ pA + pB = The above equation shows that the total final momentum of the isolated system equals its total initial momentum. This is true for elastic as well as inelastic collision. Home Next Previous

EQUILIBRIUM OF A PARTICLE
Equilibrium of a particle refers to the situation when the net external force on the particle is zero. Equilibrium of a particle requires not only translational equilibrium (zero net external force) but also rotational equilibrium (zero net external torque). According to Newton’s law, this means that, the particle is either at rest or in uniform motion. If two forces F1 and F2 act on a particle, equilibrium requires F1 = - F2 F1 F2 i.e. the two forces must be equal and opposite. or F1 + F2 = 0 F1 Equilibrium under three concurrent forces F1, F2 and F requires that the vector sum of them is zero. -F3 F1 + F2 + F3 = 0 F3 F2 The resultant of two forces F1 and F2 obtained by parallelogram law of forces must be equal and opposite to the third force F3. F2 F1 F3 By triangle law of vectors, the three sides of the triangle with arrows taken in the same order represent the equilibrium of forces. Home Next Previous

F2x + F2y + F2z are the components of F2 along x, y and z directions
A particle is in equilibrium under the action of forces F1, F2, …. Fn if they can be represented by the sides of a closed n-sided polygon with arrows directed in the same sense. O Q P R F1 F2 R = 0 F3 F1 + F2 + F3 = 0 implies that F1x + F2x + F3x = 0 F1y + F2y + F3y = 0 F1z + F2z + F3z = 0 where F1x + F1y + F1z are the components of F1 along x, y and z directions, F2x + F2y + F2z are the components of F2 along x, y and z directions and F3x + F3y + F3z are the components of F3 along x, y and z directions respectively. Home Next Previous

Gravitational Force: COMMON FORCES IN MECHANICS
The gravitational force is the force of mutual attraction between any two objects by virtue of their masses. Every object experiences this force due to every other object in the universe. All objects on the earth, experience the force of gravity due to the earth. In particular, gravity governs the motion of the moon and artificial satellites around the earth, motion of the earth and planets around the sun, and, of course, the motion of bodies falling to the earth. It plays a key role in the large-scale phenomena of the universe, such as formation and evolution of stars, galaxies and galactic clusters. The gravitational force of attraction between two objects of masses m1 and m2 held at a distance r apart is given by where G = Universal constant of gravitation. F = G m1 m2 r2 Home Next Previous

Some of the important properties of Gravitational forces:
Gravitational force is a universal force. It is pervasive. It can act at a distance without intervening medium (non-contact force). (iv) It obeys inverse square law. (v) It is always attractive in nature. (vi) It is a long range force. (vii) It is the weakest force operating in nature. (viii) It is a conservative force. Other Forces: All other forces in mechanics are contact forces. Contact force arises due to contact of one body with another (solid or fluid). Contact forces are mutual forces (for each pair of bodies) satisfying Newton’s third law. The component of contact force normal to the surfaces in contact is called ‘normal reaction’. The component of contact force parallel to the surfaces in contact is called ‘friction’. Home Next Previous

For a solid immersed in a liquid, there is an upward buoyant force equal to the weight of the liquid displaced. The viscous force, air resistance, etc. are also examples of contact forces. Weight Buoyant Force Two other common forces are tension in a string and the force due to spring. The restoring force in a string is called tension. It is customary to use the same tension T throughout the string (of negligible mass). Home Next Previous

F When a spring is compressed or extended by an external force, a restoring force is generated which is proportional to the compression or elongation (for small displacements). The spring force is written as F = - kx where x is the displacement and k is a constant. The different contact forces mentioned above arise fundamentally from electrical forces. At the microscopic level, all bodies are made of charged constituents (nuclei and electrons) and the various contact forces arising due to elasticity of bodies, molecular collisions and impacts, etc. can ultimately be traced to the electrical forces between the charged constituents of different bodies. Home Next Previous

FRICTION Static Friction F Reaction (N) (Exerted by the table)
Weight (W) (Force of gravity) fk Reaction (N) (Exerted by the table) FRICTION Static Friction The force of gravity W = mg is equal and opposite to the normal reaction force N of the table. F = 0 fs = 0 F Weight (W) (Force of gravity) fs If the body is applied a small force F, the body, in practice, does not move. If the applied force F were the only external force on the body, the body must have moved with acceleration a = F/m, however small. Clearly, the body remains at rest because some other force comes into play in the opposite direction to the applied force F, resulting in zero net force on the body. This force fs parallel to the surface of the body in contact with the table is known as frictional force, or simply ‘friction’. The subscript s stands for ‘static friction’. Static friction does not exist by itself. It comes into play the moment there is an applied force. As F increases, fs also increases, remaining equal and opposite to the applied force (up to a certain limit), keeping the body at rest. Home Next Previous

Laws of Limiting Friction:
Since the body is at rest, the friction is called ‘static friction’. Since the body is at rest upto the maximum limit (fs )max, it is called the limiting value of static friction. Static friction opposes the impending motion. The term ‘impending motion’ means motion that would take place (but does not actually take place) under the applied force, if friction were absent. When the applied force F exceeds the limit (fs )max, the body begins to move. Laws of Limiting Friction: (fs )max is independent of the area of contact. 2. It varies with the normal reaction force (N) approximately as (fs )max α N (fs )max = μs N where μs is constant of proportionality depending only on the nature of the surfaces in contact. (fs )max μs = N μs is called coefficient of static friction. The law of static friction may be written as fs ≤ μs N Home Next Previous

Sliding or Kinetic Friction
If the applied force F exceeds the limit (fs )max, the body begins to slide on the surface. When the relative motion has started, the frictional force decreases from the limiting value of static friction (fs )max. Frictional force that opposes relative motion between the surfaces in contact is called kinetic or sliding friction. It is denoted by fk. Laws of Kinetic Friction: fk is independent of the area of contact. It is nearly independent of the velocity. 3. It varies with the normal reaction force (N) approximately as fk α N or fk = μk N where μk is constant of proportionality depending only on the nature of the surfaces in contact. fk μk = N μk is called coefficient of kinetic friction. μk is less than μs. When relative motion has begun, the acceleration of the body is (F – fk) / m. For a body moving with constant velocity, F = fk. If the applied force on the body is removed, a = - fk / m and the body eventually comes to a stop. Home Next Previous

The laws of friction do not have the status of fundamental laws like those for gravitational, electric and magnetic forces. They are empirical relations that are only approximately true, but yet useful in mechanical applications. Rolling Friction A body like a ring or a sphere rolling without slipping over a horizontal plane will not suffer friction, in principle. At every instant, there is just one point of contact between the body and the plane and this point has no motion relative to the plane. In this ideal situation, kinetic or static friction is zero and the body should continue to roll with constant velocity. However, in practice, this does not happen and the body comes to rest eventually. Home Next Previous

Rolling friction has a complex origin.
To keep the body rolling, some applied force is needed. This indicates that there is a rolling friction though it is much smaller (by 2 or 3 orders of magnitude) than static or sliding friction. Rolling friction has a complex origin. During rolling, the surfaces in contact get momentarily deformed a little, and this results in a finite area (not a point) of the body being in contact with the surface. The net effect is that the component of the contact force parallel to the surface opposes the motion. Home Next Previous

Friction is a Necessary Evil
In many situations, friction has a negative role. It opposes relative motion and hence dissipates more power in the form of heat like in a machine with different moving parts. It causes wear and tear of machinery parts in contact and the tyre of automobiles. In many practical situations, however, friction is critically needed. Brakes in machines and automobiles use friction to stop the motion of them. We can not walk without sufficient friction. It is impossible for a car to move on slippery roads. On an ordinary road, the friction between the tyre and the road provides the necessary external force to accelerate the car. Home Next Previous

Methods of reducing friction
Solid (e.g. graphite), semisolid (e.g. grease) and liquid (e.g. oil) lubricants can be used to reduce the friction. Ball-bearings can be used to reduce the friction since the rolling friction is less. A thin air cushion can be used between the solid surfaces to reduce the friction. A proper combination of alloy surfaces can reduce the friction. Air Jet Lubricant Home Next Previous

CIRCULAR MOTION Bending of a cyclist   N cos N
While negotiating a curve a cyclist bends towards the centre of the curve to keep a safe balance. The angle  through which the cyclist bends can be calculated as given below:   Fcp N sin W = mg O r v The weight of the cyclist plus the cycle (W = mg) acts in the vertically downward direction. The Normal Reaction (N) acts in the direction as shown in the figure, due to bending through the angle . The centripetal force Fcp = mv2/r acts towards the centre of the curve. The Normal Reaction (N) is resolved into the components N cos and N sin as shown in the figure. N cos and mg are equal and opposite and balance each other. N sin provides the necessary centripetal force mv2/r to the cyclist. Home Next Previous

The cyclist will bend through an angle  from the vertical.
N cos = mg (1) N sin = Fcp m v2 r N sin = (2) v2 rg tan = (2) / (1)  or v2 rg  = tan-1 The cyclist will bend through an angle  from the vertical. In reality,  will be little less owing to the friction available on the road. Home Next Previous

Motion of a car on a level circular road
When a car negotiates a curve it tends to slip away from the centre of the curve and hence frictional force acts towards the centre. Static friction opposes the impending motion of the car moving away from the circle. If the road is in horizontal level the centripetal force required is given by the static friction (self adjustable) between the road and the tyres. Hence, it is the static friction that gives centripetal acceleration to the car. N v N4 N2 N1 N3 mv2/r v fs4 fs3 W = mg fs1 fs2 The forces acting on the car are: Weight of the car, mg (downwards) ii) Normal reaction, N on the car (upwards) (N = N1 + N2 + N3 + N4) iii) Static friction, fs (fs = fs1 + fs2 + fs3 + fs4) Home Next Previous

Since there is no acceleration in the vertical direction, N – mg = 0
The centripetal force is provided by the friction. m v2 r = fs (where fs = fs1 + fs2 + fs3 + fs4) For the safe turn of the car, there is a limit of magnitude of the frictional force. It can not exceed s N. i.e. fs  s N (where N = N1 + N2 + N3 + N4) m v2 r = fs  s N m v2 r  s N or m v2 r  s mg v  srg or vmax = srg or The above equation gives the maximum velocity that can be negotiated with the given value of s and r. On the other hand, s required for negotiating a curve of radius r with velocity v is given by s  v2 / rg Home Next Previous

Motion of a car on a banked circular road
N cos  The contribution of friction to the circular motion of the car can be reduced if the road is banked, i.e. raised on the outer edge of the circular road. N v N2  In the absence of friction: N1 Since there is no acceleration in the vertical direction, N sin  W = mg N cos  = mg (1) mv2/r  The centripetal force is provided by N sin . N sin  = Fcp (where N = N 1 + N 2 + N 3 + N 4) m v2 r N sin  = (2) v2 rg tan  = (2) / (1)  or v2 rg  = tan-1  is the angle through which the road is banked. The maximum velocity the car can negotiate the curve without skidding in the absence of friction is vmax =  rg tan  Home Next Previous

If the friction is available:
 N sin  N cos  N2 W = mg mv2/r N Since there is no acceleration in the vertical direction, N cos  = mg + fk sin  mg = N cos  - fk sin  (1) The centripetal force is provided by the horizontal components N sin  and fk cos . N sin  + fk cos  = Fcp fk cos  m v2 r = N sin  + fk cos  (2) fk  fk sin  But fs  s N To obtain vmax, we put fs = s N. Then Eqns. (1) and (2) become mg = N cos  - s N sin  (3) m v2max r = N sin  + s N cos  (4) cos  - s sin  mg N = From Eqn. (3), we get Home Next Previous

Substituting value of N in Eqn. (4), we get
m v2max r cos  - s sin  = mg (sin  + s cos ) or vmax = rg 1 - s tan  s + tan ) Comparing the above equation with vmax = srg we see that the maximum possible speed of a car on a banked road is greater than that on a flat (level) road. v0 =  rg tan  In the absence of friction, k= 0. At this speed, frictional force is not needed at all to provided the necessary centripetal force. Driving at this speed on a banked road will cause little wear and tear of the tyres. The above equation tell that for v < v0, frictional force will be up the slope and that a car can be parked only if tan   s. Home Next Previous

SOLVING PROBLEMS IN MECHANICS
Newton’s three laws of motion are the foundation of mechanics. More often, we may come across an assembly of different of different bodies. Each body in the assembly experiences force of gravity. While solving a problem, we can choose any part of the assembly and apply the laws of motion to that part provided we include all forces on the chosen part due to the remaining parts of the assembly. The chosen part of the assembly is called ‘system’ and the remaining part of the assembly (including other agencies of forces) as the ‘environment’. To handle a typical problem in mechanics, the following steps should be used. Draw a diagram showing schematically the various parts of the assembly of bodies, the links, supports, etc. Choose a convenient part of the assembly as one system. Draw a separate diagram which shows this system and all the forces on the system by the remaining part of the assembly. Include also the forces on the system by other agencies. Do not include the forces on the environment by the system. Such a diagram is called ‘a free-body diagram’. Home Next Previous

In a free-body diagram, include information about forces (their magnitudes and directions) that are either given or you are sure of The rest should be treated as unknowns to be determined using laws of motion. If necessary, follow the same procedure for another choice of the system. In doing so, employ Newton’s third law. i.e. if in the free-body diagram of A, the force on A due to B is shown as F, then in the free-body diagram of B, the force on B due to A should be shown as –F. Home Next Previous

Acknowledgement 1. Physics Part I for Class XI by NCERT ] Home End
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LAWS OF MOTION Aristotle’s Fallacy and Galileo’s Observations

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LAWS OF MOTION Aristotle’s Fallacy and Galileo’s Observations

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Presentation on theme: "LAWS OF MOTION Aristotle’s Fallacy and Galileo’s Observations"— Presentation transcript:

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